CALCULUS 110
APPANDIX – A
Numbers, Inequalities, and Absolute Values
Dr. Rola Asaad Hijazi
Dr. Rola A. Hijazi 06/10/1442
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Calculus is based on the real number system.
We start with the integers:
… , -3, -2, -1, 0, 1, 2, 3, 4, …
Rational Numbers
𝟏𝟐 −𝟑𝟕 𝟔𝟓 = 𝟔𝟓𝟏 𝟎. 𝟐𝟕 = 𝟏𝟎𝟎𝟐𝟕𝟑 𝟎,
𝟎 𝟎
Irrational Numbers
𝟓 𝟑 𝝅 𝐬𝐢𝐧 𝟏° 𝒍𝒐𝒈𝟏𝟎𝟐Numbers
Dr. Rola A. Hijazi
Intervals
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Inequalities
Rules for Inequalities:
Rule says that if we multiply both sides of an inequality by a negative number, then we reverse the direction of the inequality
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Dr. Rola A. Hijazi
Example 1 :
Solve the Inequality
1 + 𝑥 < 7𝑥 + 5
Solve the Inequalities
Example 2 :
4 ≤ 3𝑥 − 2 < 13
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Solve
Example 3 :
𝑥2 − 5𝑥 + 6 ≤ 0Dr. Rola A. Hijazi
Solve the Inequality
Example 4 :
𝑥3 + 3𝑥2 > 4𝑥06/10/1442 7
Exercise 25 :
𝑥 − 1 𝑥 − 2 > 0
Exercise 32 :
𝑥2 ≥ 5
Exercise 33 :
𝑥3 − 𝑥2 ≤0
Exercise 35 :
𝑥3 > 𝑥
Solve the inequality in terms of intervals and illustrate the solution set on the real number line.
Dr. Rola A. Hijazi
Absolute Value
The absolute value of a number 𝑎, denoted by | 𝑎 |, is the distance from 𝑎 to 0 on the real number line.
𝑎 ≥ 0 for every number 𝑎.
−3 = 3, 2 − 1 = 2 − 1
𝜋 − 3 = 𝜋 − 3 3 − 𝜋 = 𝜋 − 3 𝜋 − 4 = 4 − 𝜋 5 − 5 = 5 − 5
𝑎 = ቐ
𝑎 𝑖𝑓 𝑎 ≥ 0
−𝑎 𝑖𝑓 𝑎 < 0
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Distances are always
+ve
0
or
So we haveExamples
Remember that if a is negative, then –a is positive
Example5 :
Express | 3𝑥 − 2 | without using the absolute-value symbol.
Exercise 7 :
𝑥 − 2 𝑖𝑓 𝑥 < 2
Express
without using the absolute-value symbol.
It is true only when 𝑎 ≥ 0.
𝑎2 = 𝑎 if 𝑎 ≥ 0. (−𝑎)2= −𝑎 if 𝑎 < 0.
𝑎2 = 𝑎
06/10/1442 Dr. Rola A. Hijazi
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Properties of Absolute Values
Suppose 𝑎 and 𝑏 are any real numbers and 𝑛 is an integer.
Then
Suppose 𝒂 > 𝟎
𝑎𝑏 = 𝑎 𝑏 𝑎
𝑏 = 𝑎
𝑏 𝑏 ≠ 0
𝑎𝑥 = 𝑎 𝑥
𝑥 = 𝑎 ⟺ 𝑥 = ±𝑎
𝑥 < 𝑎 ⟺ −𝑎 < 𝑥 < 𝑎
𝑥 > 𝑎 ⟺ 𝑥 > 𝑎 𝑜𝑟 𝑥 < −𝑎
1 2 3
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6
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Example 6 :
| 3𝑥 − 2 | = 3 Solve
Example 7 :
| 𝑥 − 5 | < 2 Solve
Example 8 :
|3𝑥 + 2 | ≥ 2 Solve
Exercise 55 :
1 ≤ 𝑥 ≤4
4,8,10 13-22 26,28, 31, 36,44, 47-49 51-52 54
Dr. Rola A. Hijazi
Homework
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